Final answer:
The social worker should use the problem-solving approach to help the mother manage her depression and new family responsibilities by tackling problems step by step and considering available resources.
Step-by-step explanation:
The social worker's most appropriate practice approach in the case of the mother dealing with depression due to the sudden death of her sister and subsequent addition of her sister's children into her care would be C. problem-solving. This approach is integrative and pragmatic, aiming to help the client manage and cope with immediate problems by breaking them down into manageable steps, looking for viable solutions, and considering resources and support systems available. Considering her significant life stressors, including grief, increased childcare responsibilities, the need for her income, and a family history with complex challenges, the problem-solving approach would help empower the client to tackle immediate needs and plan for future goals.